Preconditioning for incompressible flows with free-surfaces and two-fluid interfaces

We investigate preconditioners for solving steady or implicit-unsteady arbitrary-Lagrangian-Eulerian moving-mesh formulations of incompressible free-surface and interfacial flow problems. The solution for the flow is obtained using the artificial compressibility method combined with a multigrid cycle. To find the surface positions in a free-surface problem, we use an iteration that is quasi-physical with the position of the free-surface driven by the kinematic condition. By modifying the fluxes through the free-surface and also using an iterative free-surface time step that is different from the flow time-step, we are able to obtain convergence rates that are almost indistinguishable from those of a flow without a free-surface. This is true in the limits of both large and small surface stresses such as gravity or surface tension and also for all flow conditions (high or low Reynolds number or small implicit Courant-Friedrichs-Lewy number). For interfacial flows, the free-surface preconditioner is extended by using a mass averaging technique. The iterative flow time-step at the interface is a mass average of the time step for the flow conditions on either side of the interface and the iterative time step for the kinematic condition is also based on mass-average quantities. We verify that this technique produces the correct results in the limits of identical fluids or a free-surface flow. We also verify that it produces well-conditioned results when the characteristic time scales of the two fluids are very different.